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We introduce a formalism to analyze partially defined functions between ordered sets. We show that our construction provides a uniform and conceptual approach to all the main definitions encountered in elementary real analysis including Dedekind cuts, limits and continuity.
Traditional examples of spaces that have an uncountable fundamental group (such as the Hawaiian earring space) are path-connected compact metric spaces with uncountably many points. We construct a compact, path-connected, locally path-connected topological space with countably many points but with an uncountable fundamental group. The construction of , which we call the “coarse Hawaiian earring” is based on the construction of the usual Hawaiian earring space where each circle is replaced with a copy of the four-point “finite circle”.
Given a dynamical system , the centralizer denotes the group of all homeomorphisms of which commute with the action of . This group is sometimes called the automorphism group of the dynamical system . We generalize the construction of Bułatek and Kwiatkowski (1992) to -Toeplitz systems by identifying a class of -Toeplitz systems that have trivial centralizers. We show that this class of -Toeplitz systems with trivial centralizers contains systems with positive topological entropy.
The Fermat numbers have many notable properties, including order universality, coprimality, and definition by a recurrence relation. We use rational points of infinite order on elliptic curves to generate sequences that are analogous to the Fermat numbers. We demonstrate that these sequences have many of the same properties as the Fermat numbers, and we discuss results about the prime factors of sequences generated by specific curves and points.
Let be a quasisimple algebraic group defined over an algebraically closed field and a Borel subgroup of acting on the nilradical of its Lie algebra via the adjoint representation. It is known that has only finitely many orbits in only five cases: when is type for , and when is type . We elaborate on this work in the case when (type ) by finding the defining equations of each orbit. We use these equations to determine the dimension of the orbits and the closure ordering on the set of orbits. The other four cases, when $G$ is type $A_n$, can be approached the same way and are treated in a separate paper.
The homophonic quotient groups for French and English (i.e., the quotient of the free group generated by the French/English alphabet determined by relations representing standard pronunciation rules) were explicitly characterized by Mestre et al. (1993). We apply the same methodology to three different language systems: German, Korean, and Turkish. We argue that our results point to some interesting differences between these three languages (or at least their current script systems).
Khan, Milićević, and Ngo evaluated the second moment of -functions associated to certain Galois orbits of primitive Dirichlet characters to modulus a large power of any fixed odd prime . Their results depend on -adic Diophantine approximation and are ineffective, in the sense of computability. We obtain an effective asymptotic for this second moment in the case of .
As shown by Cohen (1960) and Ilie and Spronk (2005), for locally compact groups and , there is a one-to-one correspondence between the completely bounded homomorphisms of their respective Fourier and Fourier–Stieltjes algebras and piecewise affine continuous maps . Using elementary arguments, we show that several (locally compact) group-theoretic properties, including amenability, are preserved by certain continuous piecewise affine maps. We discuss these results in relation to Fourier algebra homomorphisms.
It is well known that every positive integer can be expressed as a sum of nonconsecutive Fibonacci numbers provided the Fibonacci numbers satisfy for , and . For any we create a sequence called the -bin sequence with which we can define a notion of a legal decomposition for every positive integer. These sequences are not always positive linear recurrences, which have been studied in the literature, yet we prove, that like positive linear recurrences, these decompositions exist and are unique. Moreover, our main result proves that the distribution of the number of summands used in the -bin legal decompositions displays Gaussian behavior.
We give a number of results about families of Ulam sequences and sets, further exploring recent work on rigidity phenomena. For Ulam sequences, using elementary methods we give an upper bound on the density and prove regularity for various families of sequences. For Ulam sets, we consider extensions of classification work done by Kravitz and Steinerberger.
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